Question

В столярной мастерской работают мастер и его ученик. За сколько дней ученик может изготовить кресло, если он на изготовление кресла тратит на 6 дн(-ей, -я) больше, чем мастер, а работая одновременно, они могут эту работу сделать за 4 дн(-ей, -я)? Ученик может изготовить кресло за - дн(-ей, -я).

Answer 1

Вся работа - 1.

х (дн) - мастер изгот. кресло

(х+6) (дн) - ученик изгот. кресло

1/х - производительность мастера

1/(х+6) - производительность ученика

1/4 - производительность вместе

1/х+1/(х+6)=1/4       /*4x(x+6)

4(x+6)+4x=x(x+6)

4x+24+4x-x²-6x=0

x²-2x-24=0

x₁,₂=(2±√(4+96))/2=1±5

x₁=6    x₂=-4

x=6 (дн) - мастер изготовит кресло

6+6=12 (дн) - ученик изготовит кресло

Answer 2

Problem Analysis

We are given that a master and his apprentice work together in a carpentry workshop. The apprentice takes 6 days longer than the master to make a chair. However, if they work together, they can complete the chair in 4 days. We need to determine how many days it takes for the apprentice to make the chair on his own.

Solution

Let's assume that the master takes x days to make the chair. According to the given information, the apprentice takes 6 days longer than the master, so the apprentice takes (x + 6) days to make the chair.

When they work together, they can complete the chair in 4 days. This means that their combined work rate is 1 chair per 4 days. We can use the concept of work rates to set up an equation:

Work rate of the master + Work rate of the apprentice = Combined work rate

The work rate is the reciprocal of the time taken to complete a task. So, the work rate of the master is 1/x, and the work rate of the apprentice is 1/(x + 6). The combined work rate is 1/4.

Putting it all together, we have the equation:

1/x + 1/(x + 6) = 1/4

To solve this equation, we can multiply through by the common denominator, which is 4x(x + 6):

4(x + 6) + 4x = x(x + 6)

Simplifying the equation:

4x + 24 + 4x = x^2 + 6x

8x + 24 = x^2 + 6x

Rearranging the equation:

x^2 - 2x - 24 = 0

Now we can solve this quadratic equation to find the value of x.

Solving the Quadratic Equation

We can solve the quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -2, and c = -24. Substituting these values into the quadratic formula:

x = (-(-2) ± √((-2)^2 - 4(1)(-24))) / (2(1))

Simplifying:

x = (2 ± √(4 + 96)) / 2

x = (2 ± √100) / 2

x = (2 ± 10) / 2

We have two possible solutions:

x = (2 + 10) / 2 = 12 / 2 = 6

x = (2 - 10) / 2 = -8 / 2 = -4

Since the number of days cannot be negative, we discard the solution x = -4.

Therefore, the master takes 6 days to make the chair.

To find the number of days the apprentice takes, we add 6 to the master's time:

Apprentice's time = 6 + 6 = 12 days

Answer

The apprentice can make the chair in 12 days.

Sources:

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